On a final note, due to (REF ) and the origin of \(\bar{\mathcal {I}}_n\)  in large deviations for independent particles (or via variational representations as found in [1]}), one would expect that if \(F_t\in C_b(\Gamma )\)  for all \(t\in [0,T]\) , we would have for all \(n>0\)  the following representation formula for the expectation:
\( \frac{1}{n}\log \mathbb {E}\left[ e^{-n \int _0^T F_t(L_t^n)\, \mathrm {d}t}\right] = \inf _{(\mathsf {P},\mathsf {J}^+,\mathsf {J}^-)}\left\lbrace  \int _0^T \int _{\Gamma } F_t(\nu _t) \mathsf {P}_t(\mathrm {d}\nu )\, \mathrm {d}t+ \frac{1}{n}\mathcal {E}\mathrm {nt}(\mathsf {P}_0|\Pi _n) +\mathcal {I}_n(\mathsf {P},\mathsf {J}^+,\mathsf {J}^-) \right\rbrace . \) 
